i am having problem in solving equation VI given in attachment, the solution of which is also provided just below the equation. the attachment is a research paper related to sunyaev-zeldovich effect.

So, did you figure this out?

Because to me, this isn't something I would expect to solve on my own. I would make use of a table of the solutions to differential equations, or I would simply verify that this is, indeed, a solution to that differential equation.